There exists a constant $C>0$ such that, for all large $N$, if $A\subseteq \{1,\ldots,N\}$ has size at least $\frac{5}{8}N+C$ then there are distinct $a,b,c\in A$ such that $a+b,a+c,b+c\in A$.

There exists a constant $C>0$ such that, for all large $N$, if $A\subseteq \{1,\ldots,N\}$ has size at least $\frac{5}{8}N+C$ then there are distinct $a,b,c\in A$ such that $a+b,a+c,b+c\in A$.

A problem of Erdős and Sós (also earlier considered by Choi, Erdős, and Szemerédi [CES75], but Erdős had forgotten this).

This is true. The linked proof gives it in the contrapositive and with the constant cleared: every triple-free $A\subseteq\{1,\ldots,N\}$ satisfies $8\lvert A\rvert\leq 5N+C$ for a fixed $C$, for every $N$ rather than only for large $N$. It also shows the threshold is sharp, by exhibiting triple-free sets of size $(5N+16)/8$ for every $N$ divisible by $8$.